123
4.1 Ratios
Introduction
Ratio is a comparison or
relationship between two or
more quantities.
A ratio is a comparison or relationship between two or more quantities with the same unit. Therefore,
ratios are not expressed with units.
For example, if Andy (A) invested $5,000 and Barry (B) invested $4,000 in a business. The comparison
of A's investment to B's investment in the same order is called the ratio of their investments.
Expressing a Ratio of Two Quantities
When comparing two quantities, there are different ways to express the ratio. In the example above,
the ratio of A's investment to B's may be expressed in any of the following forms:
5,000 to 4,000
(separate the quantities using the word ‘to’)
5,000 : 4,000
(using a colon and read as '5,000 is to 4,000')
5,000
4,000
(as a fraction and read as '5,000 over 4,000')
In the above example, if the decimal equivalent of the fraction is used, then it must be stated as:
“A’s invesment is 1.25 times B's investment”.
Note: When representing a ratio as a fraction, if the denominator is 1, the denominator (1) must still be written.
3
For example, if the ratio of two quantities is then it is incorrect to say that the ratio is 3. It should
1
be stated as 3 or 3 : 1.
1
Expressing a Ratio of More than Two Quantities
When comparing more than two quantities, we use a colon to represent a ratio.
For example, if A's investment is $5,000, B's investment is $4,000, and C's investment is $1,000 in a
business, then the ratio of their investments is expressed as:
A : B : C = 5,000 : 4,000 : 1,000
Terms of a Ratio
The quantities in a ratio are called the terms of the ratio.
For example, the terms of the ratio 5 : 7 : 19 are 5, 7, and 19.
Equivalent Ratios
When all the terms of the ratio are multiplied by the same number or divided by the same number,
the result will be an equivalent ratio.
For example, when the terms of the ratio 12 : 15 are multiplied by 2,
we obtain an equivalent ratio of 24 : 30.
30
12 : 15
12 × 2 : 15 × 2
24 : 30
When the terms of the ratio 12 : 15 are divided by the common
factor 3, we obtain the equivalent ratio of 4 : 5.
12 : 15
12 ÷ 3 : 15 ÷ 3
15
5
24
12
4
18
9
3
12
6
2
6
3
1
0
0
0
24 : 30
4:5
24
30
=
12 : 15
=
4:5
=
12
15
=
4
5
Therefore, the ratios 12 : 15, 24 : 30, and 4 : 5 are called equivalent ratios.
4.1 Ratios
124
Example 4.1-a
Determining Equivalent Ratios
Determine whether the given pairs of ratios are equivalent.
Solution
(i)
18 : 12 and 12 : 8
(ii)
20 : 24 and 15 : 20
(i)
18 : 12
12 : 8
= 18 ÷ 6 : 12 ÷ 6
= 12 ÷ 4 : 8 ÷ 4
=3:2
=3:2
Therefore, the given pairs of ratios are equivalent.
(ii)
20 : 24
15 : 20
= 20 ÷ 4 : 24 ÷ 4
= 15 ÷ 5 : 20 ÷ 5
=5:6
=3:4
Therefore, the given pairs of ratios are not equivalent.
Reducing a Ratio to its Simplest or Lowest Terms
When two ratios are equal,
they result in the same
answer when reduced to
their lowest terms.
Comparisons are easier when ratios are reduced to their lowest terms. When all the terms of a ratio
are integers, the ratio can be reduced to its lowest terms by dividing all the terms by their common
factors.
For example, if 'A' earns $3,000, 'B' earns $4,500, and 'C' earns $6,000, then the equivalent ratio of
their earnings reduced to the lowest terms is calculated as follows:
A:B:C
A ratio is in its simplest
form when the terms do not
have a common factor other
than one.
3,000 : 4,500 : 6,000
Dividing each term by the common factor 100,
= 30 : 45 : 60
Dividing each term by the common factor 15,
=2:3:4
Now, the ratio is in its lowest terms.
By reducing the ratio to lowest terms, we can say that the earnings of A, B, and C are in the ratio of 2 : 3 : 4.
Comparing Quantities of Items That Have the Same Kind of Measure
but Different Units
When writing ratios to compare quantities of items that have the same kind of measure, the units
have to be the same.
For example, the ratio of 45 minutes to 2 hours is not 45 : 2. We are comparing ‘time’ in both these
cases; therefore, the units used have to be the same. 45 minutes : 2 hours
Converting 2 hours to minutes
using 1 hour = 60 minutes,
= 45 minutes : 120 minutes
Dividing by the common factor 15,
=3:8
Similarly, we determine the ratio of 2.5 kilometres to 3,000 metres as follows:
2.5 km : 3,000 m
Converting km to m, 2.5 km = 2,500 m,
= 2,500 m : 3,000 m
Dividing by the common factor 100,
= 25 : 30
Dividing by the common factor 5,
=5:6
Chapter 4 | Ratios and Proportions
125
Example 4.1-b
Comparing Quantities
Express each of the following ratios in its simplest form:
Solution
(i)
1.2 L to 800 mL
(ii)
16 weeks to 2 years
(i)
1.2 L to 800 mL
1.2 L : 800 mL
Converting 1.2 L to mL, 1.2 L = 1,200 mL,
= 1,200 mL : 800 mL
Dividing both terms by the common factor 400,
=3:2
(ii)
Therefore, the ratio of 1.2 L to 800 mL is 3 : 2.
16 weeks to 2 years
16 weeks : 2 years
Converting 2 years to weeks, 2 years = 104 weeks,
= 16 weeks : 104 weeks
Dividing both terms by the common factor 8,
= 2 : 13
Therefore, the ratio of 16 weeks to 2 years is 2 : 13.
Reducing Ratios When One or More of the Terms of the Ratio Are Fractions
To reduce the ratio, first convert all the terms to integers by multiplying all the terms by their lowest
common denominator, and then reduce to their lowest terms.
For example,
15 : 7 : 3
9 3
Multiplying each term by the lowest common denominator 9,
15 : 21 : 27
Dividing each term by the common factor 3,
5:7:9
Reducing Ratios When One or More of the Terms of the Ratio Are Decimal Numbers
The ratio remains the same
when all the terms are
multiplied or divided by the
same number.
To reduce the ratio, first convert all the terms to integers by moving the decimal of all the terms to the
right by the same number of places, and then reduce to their lowest terms.
For example,
2.25 : 3.5 : 5
225 : 350 : 500
Moving the decimal point of each term by 2 places to the right,
Dividing each term by the common factor 25,
9 : 14 : 20
Reducing Ratios When the Terms of the Ratio Are a Combination of
Fractions and Decimals
To reduce the ratio, first convert all the fractional terms to decimals or decimal terms to fractions,
then convert all the terms to integers. Finally, reduce to their lowest terms.
For example,
5.8 : 9 : 4
2
Multiplying each term by the common denominator 2,
11.6 : 9 : 8
Moving the decimal point of each term by 1 place to the right,
116 : 90 : 80
Dividing each term by the common factor 2,
58 : 45 : 40
4.1 Ratios
126
Reducing Ratios to an Equivalent Ratio Where the Smallest Term is 1
To make the comparison of quantities easier, we can also reduce a ratio to its equivalent ratio where
the smallest term is equal to 1, by dividing all the terms by the smallest value.
For example, if the investment amounts of 3 partners 'A', 'B', and 'C' are $35,000, $78,750, and
$59,500, respectively, then the equivalent ratio of their investments, where the smallest term is 1, is
calculated as follows:
A:B:C
35,000 : 78,750 : 59,500
1 : 2.25 : 1.7
Dividing each term by the smallest term, 35,000,
Now, the ratio is reduced to its equivalent ratio with the smallest term equal to 1.
By reducing it so that the smallest term is equal to 1, we can state: (i) B's investment is 2.25 times A's
investment and (ii) C's investment is 1.7 times A's investment.
Example 4.1-c
Reducing Ratios to Lowest Terms
Express the following ratios as equivalent ratios in their lowest whole numbers and then reduce them
to ratios where the smallest term is 1:
(ii) 2.5 : 1.75 : 0.625
(i) 2 97 : 3 31 : 5
(iii) 1.25 : 5 : 2
6
Solution
(i)
2 97 : 3 31 : 5
Converting the terms with mixed numbers to improper fractions,
25 : 10 : 5
3
9
Multiplying each term by the lowest common denominator 9,
25 : 30 : 45
Dividing each term by the common factor 5,
5:6:9
Dividing each term by the smallest term 5,
1 : 1.2 : 1.8
Therefore, 2 97 : 3 31 : 5 reduced to its lowest terms is 5 : 6 : 9 and the equivalent ratio where the
smallest term is 1 is 1 : 1.2 : 1.8.
(ii)
2.5 : 1.75 : 0.625
2.5
: 1.75 : 0.625
2,500 : 1,750 : 625
20 : 14 : 5
4 : 2.8 : 1
Moving the decimal point of each term by 3 places to the right,
Dividing each term by the common factor 125,
Dividing each term by the smallest term 5,
Therefore, 2.5 : 1.75 : 0.625 reduced to its lowest terms is 20 : 14 : 5 and the equivalent ratio where the
smallest term is 1 is 4 : 2.8 : 1.
Multiply each term by 6,
(iii) 1.25 : 5 : 2
6
7.5 : 5 : 12
Moving the decimal point of each term by 1 place to the right,
75 : 50 : 120
Dividing each term by the common factor 5,
15 : 10 : 24
Dividing each term by the smallest term 10,
Chapter 4 | Ratios and Proportions
1.5 : 1 : 2.4
5
: 2 reduced to lowest terms is 15 : 10 : 24 and the equivalent ratio where the
Therefore, 1.25 :
6
smallest term is 1 is 1.5 : 1 : 2.4.
127
Order of a Ratio
The order of presenting terms in a ratio is important. For example, if A saves $800, B saves $1,500,
and C saves $1,200, then the ratio of the savings of A, B, and C is:
1,500
A : B : C = 800 : 1,500 : 1,200 Dividing each term by the common factor 100,
= 8 : 15 : 12
In the previous example, the ratio of the savings of C, B, and A is:
1,200
800
A
1,200
C : B : A = 1,200 : 1,500 : 800 Dividing each term by the common factor 100,
= 12 : 15 : 8
B
C
1,500
800
Note that C : B : A is not the same as A : B : C.
C B A
Ratios compare the numbers in order. The ratio 12 : 15 : 8 expresses a different comparison than the
ratio 8 : 15 : 12.
Comparing Quantities
When using ratios to compare quantities of items that have different units of measure, the units of
measurement of each quantity must be included in the ratio.
For example, when baking a cake, Maggie uses 4 kilograms of flour, 2 litres of water, and 6 eggs.
Therefore, the ratio of flour to water to eggs is,
flour : water :
(kg)
(L)
eggs
(numbers)
=
4
:
2
:
6
Dividing each term by the common factor 2,
=
2
:
1
:
3
i.e., 2 kg flour : 1 litre water : 3 eggs
Rate, Unit Rate, and Unit Price
Rate
A rate is a special ratio that is used to compare two quantities or amounts that have different units of
measure. The quantities of measurements being compared are called the terms of the ratio.
For example, if a car travels 100 km using 9 L of gas, then the rate is 100 km : 9 L.
The 1st term of the ratio is measured in kilometres and the 2nd term is measured in litres.
The word ‘per’ indicates that it is a rate and it is usually denoted by a slash “/”.
Therefore, 100 km : 9 L is usually written as 100 km per 9 L or 100 km/9 L.
Rates are used in our day-to-day activities such as travelling, working, shopping, etc. For example:
travelled 90 km in 1.5 hours, worked 75 hours in 2 weeks, paid $4.80 for 3 L of milk, etc.
Example 4.1-d
Calculating Rate as a Ratio of Different Units of Measurements
A laser printer printed 88 pages in 6 minutes. Express the rate in simplified form.
Solution
The unit of the first term is in number of pages and the unit of the second term is in minutes.
Therefore, the rate of printing = 88 pages : 6 minutes (or 88 pages / 6 minutes).
Then, in simplified form = 44 pages : 3 minutes (or 44 pages / 3 minutes).
Therefore, the printing rate is 44 pages / 3 minutes.
4.1 Ratios
128
Unit Rate
If the denominator of a ratio
is 1, the 1 must be written in
the denominator.
If the denominator of a rate
is 1, we usually do not write
the 1 in the denominator.
Unit rate represents the number of units of the first quantity (or measurements) that corresponds to
one unit of the second quantity. That is, unit rate is a rate in which the rate is expressed as a quantity
which has a denominator of 1.
Rate can be converted to unit rate simply by dividing the first term by the second term.
For example,
A rate of 90 km in 1.5 hours, converted to unit rate:
= 90 km/1.5 hours
= 60 km/1 hour
= 60 km/hour
Simlarly,
A rate of 75 hours in 2 weeks, converted to unit rate:
= 75 hours/2 weeks
= 37.5 hours/1 week
= 37.5 hours/week
Example 4.1-e
Calculating Unit Rate
A car travelled 300 kilometres in 5 hours. Calculate its speed.
Solution
Distance (km) : Time (hr)
= 300 : 5
Dividing each term by 5 to reduce the second unit to 1,
= 60 : 1
Therefore, the speed of the car is 60 km per hour or 60km/hr.
Example 4.1-f
Calculating Hourly Rate of Pay
Peter worked 9 hours and earned $247.50. Calculate his hourly rate of pay.
Solution
Earnings ($) : Working Period (hrs)
= 247.50 : 9
Dividing each term by 9 to reduce the second unit to 1,
= 27.50 : 1
Therefore, his hourly rate of pay is $27.50 per hour or $27.50/hr.
Example 4.1-g
Using Unit Rates to Solve a World Problem
A car travels 90 km in 1.5 hours. At this rate, how many kilometres will it travel in 5 hours?
Solution
This can be solved by first finding the unit rate.
90 km in 1.5 hours
Therefore, the number of km per hour =
90 km
1.5 hours
= 60 km/hour
That is, the distance travelled in 1 hour = 60 km.
Therefore, the distance travelled in 5 hours = 60 × 5 = 300 km.
Chapter 4 | Ratios and Proportions
129
Example 4.1-h
Comparing Unit Rates
Car A requires 8.9 litres of gas to travel 100 km. Car B requires 45 litres of gas to travel 475 km. Which
car has the better fuel economy?
Solution
Car A : 100 km requires 8.9 litres of gas.
Therefore, the number of km per litre =
100 km
8.9 litres
= 11.24 km/litre of gas
Car B : 475km requires 45 litres of gas.
Therefore, the number of km per litre =
475 km
45 litres
= 10.56 km/litre of gas
Therefore, Car A has the better fuel economy.
Unit Price
Unit price is the unit rate when it is expressed in unit currency, dollars, cents, etc. Unit price indicates
the cost of an item for one unit of that item. The price is always the numerator and the unit is the
denominator. That is, price is expressed per quantity of 1.
Price of gas is $1.36 per litre ($1.36/litre), price of grapes is $2 per kg ($2/kg), price of juice is $0.75
per can, etc., are examples of unit price.
If the total price of a given quantity of an item is known, to find its unit price divide the total price of
the item by its quantity.
The unit price is used in comparing and making decisions in purchasing items when various options
are available. We save money when we compare the unit price of the same item in different sized
containers or different packages to determine the cheaper price per unit for our purchases.
Example 4.1-i
Calculating the Unit Price of an Item
If 3 litres of milk cost $4.80, then what is the unit price of milk?
Solution
Divide the total price of the given quantity of milk by its quantity to find the unit price of milk.
That is, $4.80 should be divided by 3 litres,
$4.80
= $1.60 per litre
3 litres
Therefore the unit price of milk is $1.60 per litre ($1.60/litre).
Example 4.1-j
Comparing Unit Prices
5 kg of almonds cost $43.50 and 4 kg of almonds cost $34.20. Which is cheaper to buy based on its unit price?
Solution
5 kg of almonds cost $43.50.
Therefore, unit price = $43.50
5 kg
= $8.70 per kg
4 kg of almonds cost $34.20
$34.20
Therefore, unit price =
4 kg
= $8.55 per kg
Therefore, based on unit price, buying 4 kg of almonds for $34.20 is cheaper than buying 5 kg of
almonds for $43.50.
Note: Unit rate and unit price problems can also be solved using the method of proportions as
demonstrated in the next section.
4.1 Ratios
130
Sharing Quantities
Sharing quantities refers to the allocation or distribution of a quantity into two or more portions (or
units) based on a given ratio.
The total of the amount shared
individually will be equal to
the original amount shared.
The ratio of the amount
shared individually, when
reduced, will be equal to the
original ratio.
For example, to allocate last year’s $1,000 profit among A, B, and C in the ratio of 2 : 3 : 5, first add
the terms of the ratio (i.e., 2, 3, and 5), which results in a total of 10 units. These 10 units represent
the total profit of $1,000, where A's share constitutes 2 units, B's 3 units, and C's 5 units, as shown in
the diagram below.
Each person's share can then be calculated, as follows:
A's share = 2 × 1,000 = $200.00
10
B's share = 3 × 1,000 = $300.00
10
C's share = 5 × 1,000 = $500.00
10
■■ The total of A, B, and C's shares will be equal to the profit amount of $1,000.
That is, the shares of A + B + C = 200 + 300 + 500 = 1,000.
■■ If we reduce the ratio of the amounts shared by A, B, and C to its lowest terms, the result would
be the original ratio. That is, 200 : 300 : 500 reduced to the lowest terms would be 2 : 3: 5.
If this year, the ratio of A's share : B's share : C's share is changed to 5 : 3 : 2 (instead of last year’s 2 :
3 : 5), and the profit amount of $1,000 remained the same, then their individual shares will change.
Their shares are recalculated as shown below:
A's share = 5 × 1,000 = $500.00
10
B's share = 3 × 1,000 = $300.00
10
C's share = 2 × 1,000 = $200.00
10
Note:
■■ The total of A, B, and C’s shares this year will be equal to the profit amount of $1,000. That is, the
shares of A + B + C = 500 + 300 + 200 = $1,000
■■ If we reduce the ratio of the amounts shared by A, B, and C to its lowest terms, the result would be
the original ratio. That is, 500 : 300 : 200 reduced to the lowest terms is 5 : 3 : 2.
Example 4.1-k
Sharing Quantities Using Ratios
A, B, and C start a business and invest $3,500, $2,100, and $2,800, respectively. After a few months
C decides to sell his share of the business to A and B. How much would A and B have to pay for C's
shares if A and B want to maintain their initial investment ratio?
Solution
Investments of A, B, and C are in the ratio of 3,500 : 2,100 : 2,800, which can be reduced to 5 : 3 : 4.
If A and B want to maintain their initial investment ratio of 5 : 3, then C’s share (of $2,800) has to be
paid for by A and B in the same ratio, 5 : 3.
By adding the ratio of A and B, we know that C’s share is to be divided into a total of 8 units, as
illustrated:
5
× 2,800.00 = $1,750.00
8
3
B would have to pay C : × 2,800.00 = $1,050.00
8
A would have to pay C :
Therefore, A would have to pay $1,750 and B would have to pay $1,050 in order to maintain their
initial investment ratio.
Chapter 4 | Ratios and Proportions
131
Example 4.1-l
Application Using Equivalent Ratios
Andrew, Barry, and Cathy invested their savings in a bank. The investments ratio of Andrew to Barry
is 2 : 3 and that of Barry to Cathy is 4 : 5. What is the investment ratio of Andrew: Barry: Cathy?
A : B = 2 : 3 and B : C = 4 : 5
Solution
Find the equivalent ratio for A : B and B : C so that the number of units in B is the same in both cases.
This can be done by finding the equivalent ratio of A : B by multiplying by 4 and that of B : C by
multiplying by 3.
A:B=2:3
Multiplying each term by 4,
= 8 : 12
B:C=4:5
Multiplying each term by 3,
= 12 : 15
Therefore, the investment ratio of Andrew : Barry : Cathy is 8 : 12 : 15.
4.1 Exercises
Answers to odd-numbered problems are available at the end of the textbook.
1. Find the ratio of the following:
a. 9 months to 2 years
b. 750 g to 3 kg
c. 30 minutes to 1 hour and 15 minutes
2. Find the ratio of the following:
a. 3 weeks to 126 days
b. 120 g to 2 kg
c. 55 minutes to 2 hours and 45 minutes
For Problems 3 to 6, express the ratios as (i) equivalent ratios in their lowest whole number and (ii) an equivalent ratio where
the smallest term is 1.
3. a. 18 : 48 : 30
b. 175 : 50 : 125
c. 0.45 : 1 : 2
4. a. 27 : 45 : 72
b. 180 : 60 : 150
c. 2.4 : 0.75
5. a.
2 1
:
3 5
b. 12 : 5 : 3
3
c. 1.7 : 8.5 : 34
3
6. a.
3 2
:
4 5
b. 65 : 91 : 13
2
c. 12 : 1.5 : 3
2
7. Which of the following ratios are equal?
a. 4 : 6 and 6 : 10
b. 8 : 10 and 28 : 35
c. 6 : 8 and 27 : 32
d. 16 : 22 and 64 : 88
c. 12 : 14 and 30 : 42
d. 12 : 26 and 30 : 65
8. Which of the following ratios are equal?
a. 16 : 20 and 24 : 30
b. 10 : 12 and 35 : 42
9. Which of the following is not an equivalent ratio of 6 : 9 : 12?
a. 4 : 6 : 8
b. 2 : 3 : 4
c. 1 : 3 : 2
d. 8 : 12 : 16
10. Which of the following is not an equivalent ratio of 16 : 24 : 12?
a. 20 : 30 : 15
b. 8 : 12 : 6
c. 28 : 42 : 21
d. 24 : 36 : 18
4.1 Ratios
132
For Problems 11 to 18, find the unit rate.
11. 525 km in 7 hours = ? km/hr
12. 680 km in 8 hours = ? km/hr
13. 154 km to 14 litres = ? km/L
14. 228 km to 19 litres = ? km/L
15. 450 words typed in 6 minutes = ? words/minute
16. 496 words typed in 8 minutes = ? words/minute
17. 261 pages in 9 minutes = ? pages/minute
18. 192 pages in 8 minutes = ? pages/minute
For Problems 19 to 24, identify the option that is less expensive based on unit rate.
19. 2 kg of flour for $3.30, or 5 kg of flour for $8.40.
20. 3 kg of sugar for $3.90, or 5 kg of sugar for $6.25.
21. 12 pencils for $4.44, or 8 pencils for $2.88.
22. 6 litres of paint for $45.60, or 5 litres of paint for $37.25.
23. 1.2 litres of juice for $2.16, or 0.8 litres of juice for $1.40.
24. 2.2 kg of jam for $11.00, or 1.5 kg of jam for $7.20.
25. In a race, participants are required to swim 3,850 metres and bike 7 kilometres. Calculate the ratio of the distance
covered by swimming to the distance covered by biking, in its lowest terms.
26. Adam, a hardware engineer, wants to install a microchip that is 38.2 mm in length into his laptop. The length of the
installation space provided in his laptop is 4.8 cm. Calculate the ratio of the length of the microchip to the installation
space, in its lowest terms.
27. An aircraft travels a distance of 3,105 km in 5 hours and 45 minutes. Calculate the ratio of the distance traveled to the
time taken, reduced to a rate of kilometres per hour.
28. Speed is defined as the ratio of the distance travelled to the time taken. If Mary, who lives in Toronto, took 6 hours
and 15 minutes to reach her parent’s home in Montreal, which is 575 km away, calculate the speed at which she was
travelling.
29. If A earns $196 for working 8 hours and B earns $98 for working 5 hours, whose average hourly rate is higher?
30. If Amanda travelled 325 km in 4 hours and 15 minutes and Ashton travelled 290 km in 3 hours and 30 minutes,
whose average speed was greater?
31. Kate earned $210 for 7.5 hours of work and Susan earned $249.75 for 9 hours of work.
a. Calculate their hourly rate.
b. Whose hourly rate was higher and by how much?
32. Jack’s monthly pay is $4,200. Steve’s weekly pay is $975.
a. Calculate their annual salary. (1 year = 12 months = 52 weeks)
b. Whose annual salary is more and by how much?
33. Emily was planning to make an authentic Indian dish for her guests. She planned to use 36 eggs, 6 litres of water, 3
tablespoons of chilli powder, and 12 tomatoes.
a. What is the ratio of the ingredients in her recipe?
b. If she decides to reduce the quantity of chilli powder to 1 12 tablespoons, calculate the new ratio of the ingredients
in her recipe.
34. In Murphy’s battery manufacturing company, 600 kg of lead, 45 kg of carbon, 30 litres of battery acid, and 120 kg of
rubber are used per day to make batteries.
a. What is the ratio of the raw materials used per day to make batteries?
b. If they alter the quantity of carbon used in the batteries and utilize 30 kg of carbon per day, calculate the new
ratio of raw materials used per day.
35. A TV cable bill of $90 is shared between two house-mates, Mike and Sarah, in the ratio of 2 : 2 12 . How much did each
person pay?
36. Alexander and Alyssa invested a total of $10,000 in a web-design business. If the ratio of their investments is 3 : 5,
what were their investments?
37. Amy and two of her friends received the first prize for a marketing case competition. They received an amount
of $7,500; however, they decided to share the prize in the ratio of the amount of time each of them spent on the
marketing case. If Amy spent 5 hours, Gary spent 8 hours, and Andrew spent 2 hours, what would be each person's
share of the prize?
Chapter 4 | Ratios and Proportions
133
38. Three friends, Andy, Berry, and Cassandra, A, B, and C, jointly insured a commercial property in the ratio of 10 : 9 : 6,
respectively. How will an annual premium of $8,000 be distributed among the three of them?
39. Three friends, Alex, Brooks, and Charlie, have decided to invest $4,000, $6,000, and $2,000, respectively, to start a
software development business. If Charlie decided to leave the business, how much would Alex and Brooks have to
pay for Charlie's share if they want to maintain their initial investment ratio?
40. Chuck decided to build a yacht with his two friends Rob and Bob and they invested $9,000, $11,000, and $6,500,
respectively. After the yacht was built, Bob decided to sell his share of the investment to Chuck and Rob. How much
would each of them have to pay if they want to maintain the same ratio of their investments in the yacht?
41. Abey and Baxter invested equal amounts of money in a business. A year later, Abey withdrew $7,500 making the ratio
of their investments 5 : 9. How much money did each of them invest in the beginning?
42. Jessica and Russel invested equal amounts to start a business. Two months later, Jessica invested an additional $3,000
in the business, making the ratio of their investments 11 : 5. How much money did each of them invest in the
beginning?
43. If A : B = 4 : 3 and B : C = 6 : 5, find A : B : C.
44. If X : Y = 5 : 2 and Y : Z = 7 : 6, find X : Y : Z.
4.2 Proportions
Proportions
When two sets of ratios are equal, we say that they are proportionate to each other. In the proportion
equation, the ratio on the left side of the equation is equal to the ratio on the right side of the equation.
Consider an example where A : B is 50 : 100 and C : D is 30 : 60.
Reducing the ratio to its lowest terms, we obtain the ratio of A : B as 1 : 2 and the ratio of C : D as 1 : 2.
Since these ratios are equal, they are equally proportionate to each other and their proportion
equation is:
A:B=C:D
The proportion equation can also be formed by representing the ratios as fractions.
Equating the fraction obtained by dividing the 1st term by the 2nd term on the left side, to the one
obtained by dividing the 1st term by the 2nd term on the right side we get:
A C
B=D
This proportion equation can be simplified by multiplying both sides of the equation by the product
of both denominators, which is B × D.
A C
B=D
Multiplying both sides by (B × D),
A
C
^B # Dh = D ^B # Dh Simplifying,
B
If two sets of fractions are
equal, then the product
obtained by crossmultiplying the fractions
will be equal.
AD = BC
The same result can be obtained by equating the product of the numerator of the 1st ratio and the
denominator of the 2nd ratio with the product of the denominator of the 1st ratio and the numerator of the
2nd ratio. This is referred to as cross-multiplication and is shown below:
A
C Cross-multiplying,
B
D
AD = BC
If 3 terms of the proportion equation are known, the 4th term can be calculated.
Therefore, A : B = C : D is equivalent to A = C .
B D
4.2 Proportions